Power saving for both supported modulus factors

The square-divisor saving pays the fixed-order divisor losses and logarithms. The original progression endpoint is absorbed under L ≤ M; this condition will be derived, not assumed, at the C.2 dyadic scale.

theorem MathlibNt.AnalyticNumberTheory.LargeSieve.LiLiuPrereqFouvry.eventually_wMaskedOriginal_zero_modulus_support_power_saving_kscale (Cscale : ℝ) (hCscale : 1 ≤ Cscale) {k : ℕ} (hk : 1 ≤ k) (j : ℕ) {η : ℝ} (hη : 0 < η) : ∀ᶠ (x : ℝ) in Filter.atTop, ∀ (M T L : ℝ), 1 ≤ M → 1 ≤ T → 1 ≤ L → M * T ≤ x → L ≤ M → ∀ (N Q : Finset ℕ), N ⊆ Finset.Ioc 0 ⌊T⌋₊ → Q ⊆ Finset.Ioc 0 ⌊L⌋₊ → ∀ (β c : ℕ → ℝ), (∀ n ∈ N, |β n| ≤ ↑((fouvryTau k) n)) → (∀ q ∈ Q, |c q| ≤ ↑((fouvryTau j) q)) → ∀ (a : ℤ), |↑a| ≤ Cscale * x → (∀ n ∈ N, β n ≠ 0 → ¬↑n ∣ a) → ∀ (P : WOriginalTuple → Prop), (∀ t ∈ wOriginalTuples N Q a, P t → x ^ η < ↑(wGCDTuple t).δ₁ ∨ x ^ η < ↑(wGCDTuple t).δ₂) → |wMaskedOriginal M N Q β c a P| + |wMaskedZeroMode M N Q β c a P| ≤ 2 * M * T ^ 2 * x ^ (-η / 4)

The same arbitrary submask is used for the original sum and its zero mode. Here T is the upper beta endpoint, and the threshold is uniform in all changing scales, residue, supports, signed coefficients, and masks.